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Year 11 Integration (Polynomials) Worksheet — Level 3 — Practice Set 2 — Answers

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Year 11 Integration (Polynomials) Worksheet — Level 3 — Practice Set 2 — Answers

Use the rule ∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C.

1.Evaluate ∫ from 2 to 3 of (15x4 + 8x + 4) dx

Answer: 657

  1. ∫ 15x4 dx = 3x5
  2. ∫ 8x dx = 4x2
  3. ∫ 4 dx = 4x
  4. F(x) = 3x5 + 4x2 + 4x
  5. F(3) − F(2) = 777 − (120) = 657
2.Evaluate ∫ from 2 to 3 of (10x4 − 4x − 2) dx

Answer: 410

  1. ∫ 10x4 dx = 2x5
  2. ∫ −4x dx = −2x2
  3. ∫ −2 dx = −2x
  4. F(x) = 2x5 − 2x2 − 2x
  5. F(3) − F(2) = 462 − (52) = 410
3.Evaluate ∫ from 0 to 1 of (8x3 + 6x2 + 2) dx

Answer: 6

  1. ∫ 8x3 dx = 2x4
  2. ∫ 6x2 dx = 2x3
  3. ∫ 2 dx = 2x
  4. F(x) = 2x4 + 2x3 + 2x
  5. F(1) − F(0) = 6 − (0) = 6
4.Find ∫ (−12x3 − 6x − 3) dx

Answer: −3x4 − 3x2 − 3x + C

  1. ∫ −12x3 dx = −3x4
  2. ∫ −6x dx = −3x2
  3. ∫ −3 dx = −3x
  4. Don't forget the constant of integration, C
5.Evaluate ∫ from 0 to 1 of (−5x4 − 12x3 − 3x2) dx

Answer: −5

  1. ∫ −5x4 dx = −x5
  2. ∫ −12x3 dx = −3x4
  3. ∫ −3x2 dx = −x3
  4. F(x) = −x5 − 3x4 − x3
  5. F(1) − F(0) = −5 − (0) = −5
6.Evaluate ∫ from 2 to 3 of (−15x4 + 8x − 2) dx

Answer: −615

  1. ∫ −15x4 dx = −3x5
  2. ∫ 8x dx = 4x2
  3. ∫ −2 dx = −2x
  4. F(x) = −3x5 + 4x2 − 2x
  5. F(3) − F(2) = −699 − (−84) = −615
7.Find ∫ (20x4 − 6x2 − 2x) dx

Answer: 4x5 − 2x3 − x2 + C

  1. ∫ 20x4 dx = 4x5
  2. ∫ −6x2 dx = −2x3
  3. ∫ −2x dx = −x2
  4. Don't forget the constant of integration, C
8.Find ∫ (20x4 + 8x3 − 2x) dx

Answer: 4x5 + 2x4 − x2 + C

  1. ∫ 20x4 dx = 4x5
  2. ∫ 8x3 dx = 2x4
  3. ∫ −2x dx = −x2
  4. Don't forget the constant of integration, C
9.Evaluate ∫ from 0 to 3 of (−12x3 + 3x2 + 3) dx

Answer: −207

  1. ∫ −12x3 dx = −3x4
  2. ∫ 3x2 dx = x3
  3. ∫ 3 dx = 3x
  4. F(x) = −3x4 + x3 + 3x
  5. F(3) − F(0) = −207 − (0) = −207
10.Find ∫ (−12x3 − 3x2 + 6x) dx

Answer: −3x4 − x3 + 3x2 + C

  1. ∫ −12x3 dx = −3x4
  2. ∫ −3x2 dx = −x3
  3. ∫ 6x dx = 3x2
  4. Don't forget the constant of integration, C
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